Let $\mathbf{p}=(p_1,\dots,p_m)$ be a vector in $[0,1]^m$ and let $\mathbf{X}=(X_1,\dots,X_m)$ be a vector of independently-distributed binomial random variables such that $X_i\sim \text{Binom}(n,p_i)$. Further, let $h(\mathbf{X})$ be a convex function on $[0,n]^m$.

Question: Is the real-valued function $g(\mathbf{p})=\mathbb{E}_\mathbf{X}[h(\mathbf{X})\ |\ \mathbf{p}]$ convex on $[0,1]^m$?

Notes:

1. This question is a follow-up from a previous question, from which we know that in the univariate case, i.e., $m=1$, $g(p)$ is convex.
2. I have done some numerical testing and it appears that $g(\mathbf{p})$ is convex
3. The function $g(\mathbf{p})$ is identical to one representation of the multivariate Berstein polynomial. I haven't seen this representation much in the literature (see another MO question). However, there is another more common representation, for which some convexity results exist.